Solving Discriminant Questions
Introduction
If you are doing A Level Maths then you have more than likely come across the quadratic formula which looks something like this:
Now it part under the square root that we are interested in b^2-4ac which is referred to as the discriminant that we are interested in.
When you sketch a quadratic curve there are three possibilities:
Graph 1: There are two points where the curve crosses the x-axis or there are two solutions. Here b^2-4ac\ge 0
Graph 2: Here the curve bounces off the x-axis so there is just one root or two equal roots. Here b^2-4ac=0
Graph 3: Here the curve does not cross the x-axis so there are no roots or no solutions to the quadratic. Here b^2-4ac\le 0
Some examples
Consider the above question. What is important here is to be able to translate the necessary English to mathematics. The key words here are equal roots. In this case b^2-4ac=0.
From the question that we have a=1,\:b\:=\:k\:and\:c\:=\:9
Substituting this into the discriminant will give us the following:
k^2-4\left(1\right)\left(9\right)=0 k^2=\:36 k=\:+6\:or\:k\:=\:-6A common mistake is to forget that there are two possible answers so do not forget the plus and minus.
No Solutions
If we consider the above question, again the key to tackling these questions is to be able to convert the English into maths. The keywords here are no real solutions. Which in terms of using the discriminant means that:
b^2-4 a c<0
Substituting:
a=3, b=-4 \text { and } c=k
into the discriminant will give the following:
\begin{aligned}
& (-4)^2-4(3)(k)<0 \\
& 16-12 k<0 \\
& 16<12 k \\
& \frac{4}{3}<k
\end{aligned}
Suppose we now consider the following question:
Looking at this for the very first time would be enough to put anyone off the question, but if you look at the function g(x) carefully you do have a quadratic. Again the way to tackle this question is to be able to translate the English text into mathematics. The main keywords here are two equal roots.
And what this means is that:
b^2-4 a c=0For this particular question:
a = 1, b = 3p and c = 14p – 3
Substituting these values into the discriminant will give us:
\begin{aligned} & (3 p)^2-4(1)(14 p-3)=0 \\ & 9 p^2-56 p+12=0 \\ & (p-6)(9 p-2)=0 \\ & p=6, p=\frac{2}{9} \end{aligned}
In terms of the value of p, we are told that p is an integer and so p = 6.
For part b of the question we simply need to substitute the value of 6 which will give us:
\begin{aligned}
& x^2+18 x+81=0 \\
& (x+9)(x+9)=0 \\
& x=-9
\end{aligned}
In this case we have a repeated root, as expected.
Summary
In this section you have met the discriminant and the important part to take away from this is recognising when to use the discriminant. The question is not going to tell you. You need to be able to recognise the keywords within the question.
The actual mathematics behind the topic is nothing terrible. It is a case of knowing the coefficients and making a suitable substitution, expanding, collecting like terms and factorising. Skills that would have been taught at GCSE and skills that you need to bring forward into A Level.
As part of your A Level Maths revision it is important that you are attempting questions such as these. You can try the questions above again or even change the wording, such as to see what happens if you have two distinct roots and how does that change your answers.
Quadratics and especially the use of the discriminant are used widely in A Level maths. Always be prepared to see a quadratic and to save time, use the discriminant to see if you are able to solve the quadratic equation that you have been given.
Whatever your goals if you need help getting those top grades then just complete the form and we will be in contact within 24 hours.